Uncategorized In the previous lecture, we had defined quantities like the first moment of an area and centroid mathematically and said that these are related to problems in mechanics. In this lecture, we are going to solve some problems using these concepts. As I had indicated towards the end of my previous lecture on this topic, the, the place where we use these concepts is where we have a distributed force. What do we mean by that? For example, if I have a beam and there is some mass on top of it so that it applies a force on the beam, the force maybe described by a function f(x) so that if I take a small section here delta x length, the force on this section, delta F is equal to f(x) times delta x. So f(x) is nothing but force per unit length. It is in dealing with such distributed forces that the concepts developed in the previous lecture are going to be handy So the question we ask is given this distribution of force f(x) what is the total force on the system and where effectively is it acting? Let me explain that a little further. The total force is going to be summation of that delta F that is acting on a small section of length delta x and this is going to be summation f(x) delta x which in the limit is going to be integration f(x) dx, that is the net force And when we say where effectively is it acting, that means what moment or torque should I apply to this beam in order to keep it in equilibrium. For example, if I have this beam fixed at this point or let me put a pin joint here, what torque should I apply here in order that this beam is in equilibrium or equivalently at which point should I apply this net force F that I have calculated above here so that the effect of this force both the torque as well as the net force is nullified? To do that, one we require that summation FY where Y is this direction be 0 and that gives me the net force F should be equal to f(x) dx. The 2nd equilibrium condition is that the torque about this pin joint vanish and that requires that the distance of the force that I am applying, call this X be such that it nullifies the torque generated by this force f(x). So F times X should be equal to summation x delta F which is nothing but integration f(x) x dx So on this beam if there is a force distribution f(x) then I have the net force F that I am supposed to apply this is about this pin joint to equilibrate the beam is going to be f(x) dx and I should have F times X where X is the distance at which the force is being applied equal to integration x f(x) dx which is nothing but the moment generated by the force distribution f(x). And therefore X equals integration x f(x) dx over F And this by definition is the definition of centroid. Therefore, the net force is the area of this force distribution curve and the point at which effectively this force acts is the centroid of this area formed by the beam and this force distribution curve That is how we use the concept of the first moment or the centroid. I must point out that when the total force capital F is applied at the centroid, no other force is needed to support the beam That is, in that situation, the force applied by the pin joint will be 0. As such, in the case of f(x) being the gravitational force, the centroid gives the position of the center of gravity. Let us take some examples Suppose I have a beam. Let us call this point x equal to 0 of length L and there is a force distribution of the form of a rectangle from point x1 to x2. In that case, one can easily see, suppose this magnitude is W, one can easily see that the net force that this applies is the area of this rectangle which is going to be x2 minus x1 times W. And where does it act? It acts at the centroid of the area formed by this force distribution and the beam And x centroid is nothing but x1 plus x2 divided by 2. Therefore I can replace this entire force or represent this entire force like this. This is a beam, this is where it is hinged. The net force is of the amount W x2 minus x1 acting at a distance this is x1, this is x2 at a distance let me write it with blue, this is x1 plus x2 divided by 2. So that is one example Next, we consider triangular loading. In that I have a beam of some length L and it is loaded like this where the maximum load is per unit length is W. Load starts at a distance x1 and goes all the way up to x2 and what we want to figure out is how much is the total force acting on the beam and where effectively is it acting? So total force is going to be the area of this triangle which is W x2 minus x1 divided by 2 and where it acts is at the centroid of this triangle Recall from the previous lecture that if I am given a triangle then with respect to one of these corners, if this distance is a and this is b, then the centroid xc is given as one third a plus b. In the present case, the 2 points are at a distance of x2 minus x1 Both the points are at a distance of x2 minus x1 from this corner. And therefore, the centroid xc is going to be at a distance from this point the hinge point here x1 plus two thirds   